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The global well-posedness and Newtonian limit for the relativistic Boltzmann equation in a periodic box

Chuqi Cao, Jing Ouyang, Yong Wang, Changguo Xiao

Abstract

In this paper, we study the Newtonian limit for relativistic Boltzmann equation in a periodic box $\mathbb{T}^3$. We first establish the global-in-time mild solutions of relativistic Boltzmann equation with uniform-in-$\mathfrak{c}$ estimates and time decay rate. Then we rigorously justify the global-in-time Newtonian limits from the relativistic Boltzmann solutions to the solution of Newtonian Boltzmann equation in $L^1_pL^{\infty}_x$. Moreover, if the initial data of Newtonian Boltzmann equation belong to $W^{1,\infty}(\mathbb{T}^3\times\mathbb{R}^3)$, based on a decomposition and $L^2-L^\infty$ argument, the global-in-time Newtonian limit is proved in $L^{\infty}_{x,p}$. The convergence rates of Newtonian limit are obtained both in $L^1_pL^{\infty}_x$ and $L^{\infty}_{x,p}$.

The global well-posedness and Newtonian limit for the relativistic Boltzmann equation in a periodic box

Abstract

In this paper, we study the Newtonian limit for relativistic Boltzmann equation in a periodic box . We first establish the global-in-time mild solutions of relativistic Boltzmann equation with uniform-in- estimates and time decay rate. Then we rigorously justify the global-in-time Newtonian limits from the relativistic Boltzmann solutions to the solution of Newtonian Boltzmann equation in . Moreover, if the initial data of Newtonian Boltzmann equation belong to , based on a decomposition and argument, the global-in-time Newtonian limit is proved in . The convergence rates of Newtonian limit are obtained both in and .
Paper Structure (22 sections, 19 theorems, 445 equations)

This paper contains 22 sections, 19 theorems, 445 equations.

Table of Contents

  1. Introduction
  2. The relativistic Boltzmann equation
  3. A brief history on the Newtonian limit of relativistic Boltzmann equation
  4. Main results
  5. Key points of the proof
  6. Organization of the paper
  7. Notations
  8. Preliminaries
  9. Bessel function
  10. Newtonian Boltzmann collision operators
  11. Hilbert-Schmidt formulation
  12. Glassey-Strauss expression
  13. Some auxiliary estimates
  14. Nonlinear estimate
  15. Global existence for the relativistic Boltzmann equation
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\beta>4$ and $(M_{\mathfrak{c}},\mathbf{J}_{\mathfrak{c}},E_{\mathfrak{c}})=(0,\mathbf{0},0)$. There exist constants $\varepsilon_{0}>0$, $C_{0}>0$ and $\lambda_{0}>0$, which are all independent of the light speed $\mathfrak{c}\ (\gg1)$, such that if $F_{0,\mathfrak{c}}(x, p)=J_{\mathfrak{c}}(p then the relativistic Boltzmann equation 1.3-0-1.4-0 admits a unique global solution $F_{\mathfrak{

Theorems & Definitions (33)

  • Theorem 1.1: Global existence
  • Lemma 1.2
  • Theorem 1.3: Newtonian limit in $L_p^1 L_x^{\infty}$
  • Remark 1.4
  • Theorem 1.5: Newtonian limit in $L^{\infty}_{x,p}$
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 23 more